\section{Orbital perturbations}
\begin{frame}{\thesection.\ \insertsection}
The Keplerian two-body orbit is very much an idealized motion.
\begin{itemize}
    \item It works well for short periods of time, but there are several factors that cause the actual motion to deviate from the Keplerian orbit.
\end{itemize}
\begin{center}\includegraphics{fig_5_p53.pdf}\end{center}
\end{frame}

\begin{frame}[t]{\thesection.\ \insertsection}
These factors include:
\begin{enumerate}
\item Non-sphericity of the primary \( m_1 \) (the Earth is not perfectly spherical)
    \begin{itemize}
    \item In the Keplerian orbit, masses \( m_1 \) and \( m_2 \) are assumed to be point masses.
        \begin{itemize}
        \item[\mysquare] If the primary \( m_1 \) is spherically symmetric, then it can be treated as a point mass.
        \item[\mysquare] Due to the small size of $m_2$ (compared with the distance from the center of $m_1$),
            this is reasonable approximation for \( m_2 \).
        \end{itemize}
    \item As we shall see later, the non-sphericity of the primary must be taken into account.
    \end{itemize}
\item Presence of other bodies and their gravitational fields
    \begin{itemize}
    \item For a spacecraft orbiting the Earth,
        the spacecraft is also influence by the sun and moon
        (and to a much lesser extent, the other planets).
    \end{itemize}
\item Atmospheric drag
    \begin{itemize}
    \item For near-Earth orbits, there is still some residual atmosphere,
        creating drag on the spacecraft. This results in a gradual orbit decay.
    \end{itemize}
\item Solar radiation pressure
    \begin{itemize}
    \item Light from the sun (photons) creates pressure on the lit surface of the spacecraft,
        which is caused by momentum transfer from the photons to the spacecraft surface.
    \end{itemize}
\end{enumerate}
\end{frame}

\begin{frame}{\thesection.\ \insertsection}
These effects are perturbations to the Keplerian orbit. \\
Let $\vec f_p$ be the perturbative acceleration due to the perturbing effects. \\
Then the true equation of motion is
\[ \ddot{\vec{_{\,}r}} = -\frac{\mu}{r^3} \vec{_{}r} + \vec{f}_p \]
There are two approaches for dealing with the perturbations:
\begin{enumerate}
\item Special perturbations
    \begin{itemize}
    \item[\mysquare] Determine the effects numerically, by performing some kind of numerical integration.
    \item[\mysquare] They are called ``special'', since the solution is only valid for one set of initial conditions.
    \end{itemize}
\item General perturbations
    \begin{itemize}
    \item[\mysquare] Determine the effects analytically, and expressions for
        \[\frac{da}{dt}, \frac{de}{dt}, \frac{di}{dt}, \frac{d\Omega}{dt}, \frac{d\omega}{dt}\]
        are obtained.
    \item[\mysquare] Approximations are often made in their derivations, so they are not as accurate as special one.
    \end{itemize}
\end{enumerate}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Special perturbations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\vspace{-3pt}
1. Cowell' Method (In the early 20th century)
\begin{itemize}
\item Cowell’s method numerically integrates the equations of motion directly.
\item We represent all quantities in an inertial frame of reference \( \mathcal{F}_I \):
    \[
        \vec{_{}r} = \vec{\mathcal F}^T_I r, \quad
        \vec{v} = \vec{\mathcal F}^T_I v, \quad
        \vec{f}_p = \vec{\mathcal F}^T_I f_p, \quad
        \vec{_{}r}_0 = \vec{\mathcal F}^T_I r_0, \quad
        \vec{v}_0 = \vec{\mathcal F}^T_I v_0
    \]
\item \textcolor{blue}{We can write the equations of motion in the first order form appropriate for numerical integration, namely}
    \vspace{-6pt}
    \[\textcolor{blue}{
        \begin{bmatrix} \dot{r} \\ \dot{v} \end{bmatrix} =
        \begin{bmatrix} v \\ -\frac{\mu}{(r^\text Tr)^\frac{3}{2}}r + f_p \end{bmatrix},
        \begin{bmatrix} r(0) \\ v(0) \end{bmatrix}=
        \begin{bmatrix} r_0 \\  v_0 \end{bmatrix}
    }\]
\end{itemize}
\vspace{-12pt}
Remarks:
\begin{itemize}
    \item To achieve the required accuracy, small time-steps are typically required, making it computationally expensive.
    \item The round-off errors due to numerical integration can accumulate quite rapidly, making long term solutions inaccurate.
    \item This is less of an issue today than it was in the past, due to the computational power and sophisticated techniques available.
\end{itemize}
\end{frame}

\begin{frame}[t]{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
2. Encke’s Method (1857)
\vfill
This method works by numerically integrating the deviation of the true (perturbed) orbit satisfying  
\[
    \ddot{\vec{_{\,}r}} = -\frac{\mu}{r^3} \vec{_{}r} + \vec{f}_p, \quad
    \vec{_{}r}(0) = \vec{_{}r}_0, \quad \dot{\vec{_{\,}r}}(0) = \vec{v}_0
\]
from a reference Keplerian orbit  
\[
    \ddot{\vec{\rho}} = -\frac{\mu}{\rho^3} \vec{\rho}, \quad
    \vec{\rho}(0) = \vec{_{}r}_0, \quad
    \dot{\vec{\rho}}(0) = \vec{v}_0
\]
which can be determined analytically using the two-body solution.
Define the deviation from the Keplerian orbit as
\[\delta \vec{_{}r} \triangleq \vec{_{}r} - \vec{\rho}\]
\textcolor{blue}{Taking the difference, we have}
\[\textcolor{blue}{
    \delta\!\ddot{\vec{_{\,}r}} = -\frac{\mu}{\rho^3} \left[ \delta \vec{_{}r} -
    \left( 1 - \frac{\rho^3}{r^3} \right) \vec{_{}r} \right] + \vec{f}_p, \quad
    \delta \vec{_{}r}(0) = \vec{0}, \quad \delta\!\dot{\vec{_{\,}r}}(0) = \vec{0}
}\]
\[\textcolor{blue}{\vec{_{}r} = \vec{\rho} + \delta \vec{_{}r}}\]

\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
Remarks:
\begin{itemize}
    \item It has been found to be up to ten times faster than Cowell’s method.
\begin{itemize}
    \item [\scalebox{0.6}{$\blacksquare$}]Since the quantity \(\delta \vec{_{}r}\) changes much more slowly than \(\vec{_{}r}\), the numerical integration scheme can use much larger time-steps for the same accuracy.
\end{itemize}
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{General perturbations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
It obtains analytical expressions for the changes in the orbital elements.
\vfill
We will use the cylindrical coordinate frame \( \mathcal{F}_o \), which has basis vectors \( \vec{x}_o, \vec{y}_o, \) and \( \vec{z}_o \).
\begin{center}\includegraphics{fig_5_1.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.1:} Cylindrical coordinate frame\end{center}
\vspace{12pt}
Let us express the perturbing force in \( \mathcal{F}_o \) coordinates as
\[\vec{f}_p = f_r \vec{x}_o + f_\theta \vec{y}_o + f_z \vec{z}_o\]
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
We can obtain variational equations for the orbital elements:
\begin{align*}
\frac{da}{dt} &= \frac{2a^2}{\sqrt{\mu a(1 - e^2)}} [e \sin \theta f_r + (1 + e \cos \theta) f_\theta] \\
\frac{de}{dt} &= \sqrt{\frac{a(1 - e^2)}{\mu}} \left[\sin \theta f_r + \frac{2 \cos \theta + e(1 + \cos^2 \theta)}{1 + e \cos \theta} f_\theta\right] \\
\frac{di}{dt} &= \sqrt{\frac{a(1 - e^2)}{\mu}} \frac{\cos(\omega + \theta)}{1 + e \cos \theta} f_z \\
\frac{d\Omega}{dt} &= \sqrt{\frac{a(1 - e^2)}{\mu}} \frac{\sin(\omega + \theta)}{\sin i(1 + e \cos \theta)} f_z \\
\frac{d\omega}{dt} &= \sqrt{\frac{a(1 - e^2)}{\mu}} \left[ -\frac{\cos \theta}{e} + \frac{(2 + e \cos \theta) \sin \theta}{e(1 + e \cos \theta)} f_\theta - \frac{\sin(\omega + \theta)}{\tan i(1 + e \cos \theta)} f_z \right]
\end{align*}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Gravitational perturbations due to a non-spherical primary body}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
Now that the perturbation methods have been formulated, we need expressions for the perturbative accelerations \( \vec{f}_p \).
\vspace{12pt}
\begin{itemize} \setlength{\itemsep}{10pt}
    \item We will focus on the effects due to a non-spherical primary, which are very important for Earth-orbiting satellites.
    \item \textcolor{blue}{The Earth's shape is oblate, which is flattened at the poles.}
\end{itemize}
\begin{center}\includegraphics{fig_5_2.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.2:} Earth flattering at the poles\end{center}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
We can write the perturbative force per unit mass in cylindrical coordinates.
\[ \vec{f}_p = f_r \vec{x}_o + f_\theta \vec{y}_o + f_z \vec{z}_o \]
where
\begin{align*}
f_r &= \frac{3\mu J_2 R_e^2}{2r^4} (3\sin^2 i \sin^2 (\omega + \theta) - 1) \\
f_\theta &= \frac{3\mu J_2 R_e^2}{2r^4} \sin^2 i \sin (2(\omega + \theta)) \\
f_z &= \frac{3\mu J_2 R_e^2}{2r^4} \sin 2i \sin (\omega + \theta) \\
J_2 &= 1.083 \times 10^{-3}
\end{align*}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Effect of $J_2$ on the orbital elements}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
Now that we have an expression for the perturbative force per unit mass due to \( J_2 \), we can use the general perturbation method to determine the effect.
\begin{itemize}
    \item We can obtain the average rates of change of the orbital elements:
\end{itemize}
\begin{align*}
&< \dot{\Omega} > = -\frac{3J_2R_e^2}{2(1 - e^2)^2} \sqrt{\frac{\mu}{a^7}} \cos i \\
&< \dot{\omega} > = \frac{3J_2R_e^2}{4(1 - e^2)^2} \sqrt{\frac{\mu}{a^7}} (5 \cos^2 i - 1) \\
&< \dot{a} > = < \dot{e} > = < \dot{i} > = 0
\end{align*}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
The oblateness effects of the earth (through \( J_2 \)) only affect \(\Omega\) and \(\omega\) in the long-term.
\begin{itemize}
    \item The orbital plane rotates about the Earth's spin axis at an average rate of $<\dot{\Omega}>$.
    \item The argument of perigee rotates about the orbit normal at an average rate of $<\dot{\omega}>$.
\end{itemize}
\begin{center}\includegraphics{fig_2_8.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.3:} Orbital elements\end{center}
\end{frame}

\begin{frame}

\begin{center}
\large References
\end{center}

\begin{description}
\item[{[1]}]  A. H. J. de Ruiter, C. J. Damaren, J. R. Forbes, Spacecraft Dynamics and Control, an Introduction, John Wiley \& Sons Ltd, 2013.
\item[{[2]}] F. L. Markley, J. L. Crassidis, Fundamentals of Spacecraft Attitude Determination and Control, Springer, 2014.
\item[{[3]}] L. Mazzini, Flexible Spacecraft Dynamics, Control and Guidance, Springer, 2016.
\item[{[4]}] V. A. Chobotov, Spacecraft Attitude Dynamics and Control, Krieger Publishing Company, 1991.
\item[{[5]}] M. J. Sidi, Spacecraft Dynamics and Control, Cambridge University Press, 1997.
\end{description}
\end{frame}
